As we we march down this road of suffering toward death, it is only the one who has lost all perspective that gives any serious thought to his hair style or his manner of dress, or judges anyone by reference to cosmetic criteria.

Abstract Here it is proposed -with reference to Isaac Newton- that the apparent conflict between the strictly monotheistic Jewish and the Trinitarian Christian view of Jesus if Nazareth is based on a misunderstanding, a misunderstanding arising from the fact that G-d is outside of time and space, while any incarnate soul is ensconced within it.

As a fellow of Trinity College, Cambridge, Isaac Newton was required to take holy orders, and to become of minister in the Church of England. This involved taking a vow of celibacy, and recognizing the Thirty-Nine Articles of the Church of England. Apparently Newton was desperate to avoid this -and ultimately did avoid it- in no small part because of his disbelief in one of the most fundamental doctrines of Christianity, the doctrine of the Trinity. Like a Christian, Newton held that Jesus of Nazareth was the son of G-d, but that like a Jew, he held G-d to be a solitary, indivisible being, and thus that Jesus was less than G-d. Newton’s views on this subject are summarized in an unpublished document from 1673 (1):

1. The word G-d is nowhere in the scriptures used to signify more than one of the three persons at once.

2. The word G-d put absolutely without restriction to the Son or Holy Ghost doth always signify the Father from one end of the scriptures to the other.

3.Whenever it is said in the scriptures that there is but one G-d, it is meant the Father.

4. When, after some heretics had taken Christ for a mere man and others for the supreme G-d, St John in his Gospel endeavoured to state his nature so that men might have from thence a right apprehension of him and avoid those heresies and to that end calls him the word or logos: we must suppose that he intended that term in the sense that it was taken in the world before he used it when in like manner applied to an intelligent being. For if the Apostles had not used words as they found them how could they expect to have been rightly understood. Now the term logos before St John wrote, was generally used in the sense of the Platonists, when applied to an intelligent being and the Arians understood it in the same sense, and therefore theirs is the true sense of St John.

5. The Son in several places confesseth his dependence on the will of the Father.

6. The Son confesseth the Father greater, then calls him his G-d etc.

7. The Son acknowledgeth the original prescience of all future things to be in the Father only.

8. There is nowhere mention of a human soul in our Saviour besides the word, by the meditation of which the word should be incarnate. But the word itself was made flesh and took upon him the form of a servant.

9. It was the son of G-d which He sent into the world and not a human soul that suffered for us. If there had been such a human soul in our Saviour, it would have been a thing of too great consequence to have been wholly omitted by the Apostles.

10. It is a proper epithet of the Father to be called almighty. For by G-d almighty we always understand the Father. Yet this is not to limit the power of the Son. For he doth whatsoever he seeth the Father do; but to acknowledge that all power is originally in the Father and that the Son hath power in him but what he derives fro the Father, for he professes that of himself he can do nothing.

11. The Son in all things submits his will to the will of the Father, which could be unreasonable if he were equal to the Father.

12 The union between him and the Father he interprets to be like that of the saints with one another. That is in agreement of will and counsel.

As a biblical scholar Newton was second to none, but there is a subtly here of which he was -because of his insufficiently sophisticated view of view of time and space- unaware. This can be best expressed by reference to the form of Jewish mysticism known as “Kabbalah” (2). In Kabbalah there are 5 levels of awareness or G-d-Consciousness – from bottom to top, Nefesh, Ruach, Neshama, Chaya and yechida. Kabbalah Online:

Regarding these levels of soul, the Zohar states that when a person is born, he is given a Nefesh from the world of Asiya, the lowest world, representing the greatest concealment of G‑d. If, through his divine service and proper action, he makes himself worthy, he is then given Ruach on the plane of Yetzira. With greater effort, he can earn the revelation of Neshama, paralleling the world of Beriya. If he purifies himself greatly, he may be able to attain the level of Chaya paralleling Atzilut, and even yechida – the G‑d consciousness of the level of Adam Kadmon and beyond. (“Beyond,” because the level of soul called yechida in essence transcends all the worlds, since it is never separated from G‑d. It is described as being “truly part of G‑d above,” and as “a spark of the Creator enclothed within a spark of the created”).

When a soul has ascended to the level of yechida, it is no longer separate from G-d. Kabbalah takes the Mans’s-Eye-View, from which the fallen, disgraced soul, aspires to ascend to the level of G-d, but there is another G-d’s Eye View from which the perfect soul knowingly descends to the level of man. Whilst he was incarnated in a human body, it is inevitable that Jesus soul lacked the state of yechida, but he could recollect possessing this state and declared himself to be returning to such a state (John 16:28: “”I came forth from the Father and have come into the world; I am leaving the world again and going to the Father.)” Was/is Jesus G-d? No and yes. Incarnated in a human body, he was as Newton shows in his twelve points, distinct from G-d and subordinate to him. This is the only way to make sense of his desire not to be crucified and his plea to G-d to save him from it (Mathew 26: 39 “…he fell with his face to the ground and prayed, “My Father, if it is possible, may this cup be taken from me. Yet not as I will, but as you will.”). But he possesses another existence in which he is one with G-d. This approach does not not conflict with the Jewish idea of the singular nature of G-d, a conflict that drove Newton to Arianism (Arianism = a Christological concept – attributed to Arius (c. AD 256 – 336), a Christian presbyter in Alexandria, Egypt- which asserts the belief that Jesus of Nazareth is the Son of G-d who was begotten by G-d the Father at a point in time, and is distinct from the Father and is therefore subordinate to the Father), and nor does it conflict with the Trinitarian idea that the soul of Jesus alone has the capacity to be both incarnated and one G-d. The trick is to realize -as the Kabbalists do- that the same soul can inhabit distinct domains, the finite, limited domain of man, and the eternal, unlimited domain of G-d, known in Kabbaha as Ein Sof. Of course, every soul incarnated in the former domain is distinct from G-d and subordinate to him, but this is not to say that a soul that is incarnated in former situation cannot in principle be one with G-d in the latter, and thus not subordinate to him. But for the possibility of such a dual identity, the finite world cannot be -as it is- an emanation of the infinite.

Abstract In the world of cryptocurrency, there is a well-known limitation on the maximum share of the computing power of the bitcoin network that any one set of computers in the network can possess before it ceases to be decentralized: if any one group of users were to control more than 50% of the network’s mining hashrate, then the possibility exists that they could set up a financial dictatorship, thereby destroying the feature of cryptocurrencies that make them so attractive to those that oppose the inequitable distribution of wealth and power – decentralization. Here the idea is tabled that there is a similar decentralization-born limitation on the building of scalable quantum computers. It is argued that this arises from the the maximum share of the computing power of the “quantum network” that any one set of classically instantiated devices can possess, a share limited by the value = 1/2 and by the Riemann Hypothesis.

PART 1

Cryptography

Wikipedia defines cryptography like this:

Cryptography… is the practice and study of techniques for secure communication in the presence of third parties (called adversaries).

For centuries, the key used to code a message and the key used to decode it were symmetric, i.e. the means of decoding the message was a matter of reversing the means of encoding the message. Metaphorically speaking, messages were placed in locked boxes (encoded) by one party (Alice), sent to another party (Bob), and opened by a single key (code) that must also exchanged, and could fall into the hands of an adversarial third party (Eve) en route. But with the advent of public key cryptography -invented in 1976 by mathematicians Whitfield Diffie and Martin Hellman (1)- an asymmetric encoding/decoding process was introduced. Metaphorically, Alice puts her message in a box, locks it with a key that she alone has, and sends it to Bob. Because he doesn’t have the key, Bob can’t open the box, but he can put a further lock on it, which he does, and sends it back to Alice. Alice then removes her lock and re-sends the box to Bob, who removes his lock and reads Alice’s message.

Trapdoor One-Way Functions

This more sophisticated form of code-making depends on something known as a ‘trap-door one-way-function’. A one way function is a mathematical operation that it easy to perform in one direction but hard to perform the other without a trapdoor. The simplest example of a one-way function is multiplication/division where this concerns prime numbers. A prime number is indivisible by any number other than 1 and itself, and a composite is the product of primes. 2 for example is prime, 3 is prime, but 4 is the product of 2 and 2. 5 is prime, but 6 is the product of 2 and 3… and so on. It is easy to multiply primes to create a composite, but difficult to divide a composite into its constituent primes. This division process is called factoring, and although it is easy enough to factor 15 (3.5) or 20 (2 .2 .5), imagine trying to factor the 200 digit integer

It takes a pocket calculator a spit second to do the multiplication, but it would take a single 2.2 GHz computer roughly 75 years to do the division.

Elliptic Curve Cryptography

There is an ancient question of whether there an integer that is the area of a right triangle whose sides have rational length. Such integers are called ‘congruent’. 6 for example is a congruent number because there is a right triangle whose area is 6 and whose sides are 3, 4 and 5. The problem of determining if a is the area of a right triangle having rational sides is equivalent to he problem of finding rational solutions for x and y to the cubic equation

which is in turn equivalent to the problem of determining if there are infinite rational points on an elliptic curve, the curve underlying Andrew Wiles’ proof of Fermat’s Last Theorem (2). Another form of trapdoor one-way function, and that used by Bitcoin and crytocurrencies, arises in the context of elliptic curves. The operation of addition can be defined on an elliptic curve by appeal to the fact that if we draw a line through two points P and Q on such a curve, the line intersects the curve at a third point, and since the elliptic curve is symmetric about the x-axis gives us a further point which is the reflection of this point We call this reflected point P + Q.

Multiplication can be defined by drawing a tangent line through a point P on an elliptic curve to arrive at another reflected point that we call P+P or 2P:

By drawing a tangent line through this point at 2P we create a further point that we call 3P. From here we can simply go on drawing lines and creating further points – 2P, 3P, 4P… and so on:

Now while it is easy, given a base point P, to work out how to produce n . P, it is not easy to go in the reverse direction: given n . P, what was the base point P?

BQP = NP

Problems that can be solved by computers in an efficient amount of time (where efficient means that as the problem-size grows arithmetically, the number of steps required to solve the problem grows “polynomially “rather than exponentially): problems whose solutions can be arrived at in an efficient amount of time we classify as “P” for polynomial time, problems whose solution can be verified in an efficient amount of time we classify as NP for non deterministic polynomial time; all problems in P are also in NP, but some NP problems- NP-hard- are not known are in P. NP-Complete problems are as hard as any problem in NP, and the classic example is The Travelling Salesman, which involves a salesman who must, after starting from a home-city, visit a number of cities exactly once before returning the home city for a certain cost. To see that there is no efficient way to solve this problem using a classical computer, let the home city = the halt state of a computer, let every other city = an atomic instruction of a computers program. If an instruction can be executed, assign a cost of 1, if an instruction cannot be executed, assign a cost of 2. If and only if the salesman can complete a circuit that visits every city exactly once for a cost of the number of cities, there is some computer that will halt when run with some input. If the number of cities is the same as the number of instructions in the program of the machine evaluating the TSP instance, the evaluation is a self-evaluation.

And if we assign a cost of 2, rather than 1, then this machine is required to determine that an input can only be run by a more complex program than itself.

This proves that there are some problems that classical computers, no matter how powerful, cannot solve in an efficient amount of time. It also proves that the problem of whether there are some problems that a classical computer can’t solve in an efficient amount of time (called “P versus NP”) cannot be solved in any amount of time by a classical computer, and yet you and I have just solved it, which implies at least that human mind is -or has access to- a non-classical computational intelligence. Every Travelling Salesman problem can be transformed into a Factoring Problem by identifying TSP problems with integers based on the number of cities in the problem = the number instructions in a computers program. Factoring expressed in terms of whether a number has a prime factor not greater than n is equivalent to Travelling Salesman expressed in terms of whether there is cycle whose cost is not greater than n, and a classical computer can’t factor a number n such that n is greater than the number of instructions in its own program… But Shor’s Algorithm (3) for a quantum computer permits the factoring of integers and solving the discrete logarithm problem in a efficient amount of time, from which it follows that the limit there is on classical computers in virtue of which they can’t solve certain problems in an efficient amount of time, nor solve certain problems in any amount of time, is inapplicable to quantum computers. It follows that the human mind is, not simply a classical computer, but has quantum computational elements. BQP is the class of problems efficiently solvable by a classical computer, and while P!=NP, BQP = NP. Since factoring and the discrete logarithm problem are both in NP, it follows that a quantum computer can solve them efficiently, and that if a scalable quantum computer were built, any information protected by the RSA cryptosystem or by elliptic curve cryptography would thereby by be rendered insecure. More generally a scalable quantum computer can reverse the arrow of time…

PART II

The Golden Key

Called by John Derbyshire “The Golden Key” (4), Euler’s classic argument (5) establishes that the product continued to infinity of this fraction

in which the numerators are prime numbers and the denominators are one less than the numerators, equals the sum of the infinite series

and they are both infinite. To prove his point to Euler invites us to imagine the extraction from the second series a prime denominator and all remaining multiples of that prime denominator until everything except the first term 1 has been eliminated. Let

Then

This leaves

To eliminate the denominators that are divisible by 3, we divide both sides to get

Subtracting again eliminates all remaining denominators that are multiples of 3 leaving

Applying this eliminatory process to all of the prime numbers leaves

This is a thought-experiment -mere imagination- but if these eliminations could be performed in the physical world, they would result in the disappearance of any distinction between the form and the content of a coordinate system, and therefore the shrinking of space and the slowing of time to a zero-dimensional point. With all of reality contracted to a zero-dimensional point, the distinction between the world and the mind that surveys it is lost. This is the singularity at root of general relativity. It is not -as is often maintained- all of the mass in the universe compressed down to a point, an infinitely heavy object. This is a gross misconception resulting from the assumption of atomism, and the prioritization of space over light, a misconception responsible for all that is ugly in contemporary physics, including black holes as infinite at their centers (6), the incongruity of relativity and quantum mechanics (7), the infinities of quantum field theory (8), and the flat rotation curves of distant galaxies that seem to call for dark matter (9, 10)… to name a handful of significantly troublesome things. Rather, this singular state is a state such that all of the space and time -all the gaps and holes- in the universe are from the perspective of this singularity excised, leaving only infinitely concentrated light, an infinitely light object in both senses of the term “light”. On this account there are two forms of curvature resulting either from an imbalance of light and space in favor of light (the most extreme form of this imbalance is the origin of the universe) or from an imbalance of light and space in favour of space (the most extreme forms of this imbalance are black holes).

The first idea we can take from Euler’s thought-experiment is that, since both prime-density and energy-density must at this point be infinite, the spatio-temporal development of the universe from a central singular point towards an ever-increasing state of decentralization is a process involving the distribution of the prime numbers. We can add to this that this process of decentralization takes place according to something known as the Riemann Hypothesis (11), which says that the thinning of primes -the spreading of prime-energy over time and space- with arithmetic increase cannot exceed the upper and lower bounds such as those marked in red and blue in the graphs below:

The Generalized Riemann Hypothesis (12, 13) extends the Riemann Hypothesis by reference arithmetic progressions associated with the equation q n + a where q and n have no common factor greater than 1. In a universe whose fundamental condition is an infinite state of prime and energy density which is diffused from the point of view of any and every frame of reference according to the Generalized Riemann Hypothesis, time has a forwards direction associated with the loss of prime and energy density and a backwards direction associated with a gain in prime and energy density. Because the loss of prime-density predominates over any gains in prime-density, the direction of time is given by the GRH, a balance of the two forms of curvature is maintained by the equality

The waveform () we obtain by taking difference between the number of primes not greater than x and an approximating formula such as

and dividing by x is a superposition of smooth waves:

Or from another perspective on the same phenomenon, we can obtain the superposition by taking the difference between the primes themselves and an approximating formula such as

The intimate connection there is between number theory and quantum field theory can be simply illustrated (14) by associating the creation operators and to the prime numbers … Now we have identified the unique ‘factorization’ of a state into creation operators acting on the ‘vacuum’ with the unique factorization of an integer into prime numbers (and we have a hierarchy of states: |1> is the ‘vacuum’; |2> and |3> and |5> are one-particle states; |6> is a two-particle state… and so on). By reference to the Witten index (15) -the number of bosonic minus the number of fermionic zero-energy states- we see that the Mobius inversion function

={1 = n has an even number of distinct factors, -1 = n has an odd number of distinct factors, 0 = n has a repeated factor}

is equivalent to the operator that distinguishes bosonic from fermionic states, with = 0 when n has a repeated factor being equivalent to the Pauli exclusion principle. If we re-express the Mertens function (which sums the 1s and -1s of the Mobius function) as , we see that sums of these states give us essentially the same composite spiral-wave as before.

Assuming that there are an equal number of non-zero-energy bosonic and fermionic states, this wave depicts the zero-energy fluctuations of these particles, the energy fluctuations of the vacuum. This is to say that the vacuum is the basis of everything -everything emanates from the vacuum- and that the vacuum is far from vacuous. There is here a distinction between symmetric superpositions such as (where the associated progression is arithmetically continuous) and asymmetric superpositions familiar from quantum mechanics (where the associated progression is arithmetically discontinuous). These examples concern the case where q =1 and a = 0, but we can easily construct approximating formulas and wave-forms for all the other possible values of q and a. The value , associated as it is with a very particular balance of prime-density and sparsity, signifies the dividing line between classical objects that are constrained to travel in time in an arithmetic manner, and quantum objects that transcend arithmetic and are not so constrained. A key aspect of this mathematical scheme is that we can take any arithmetic progression associated to q n + a and to an L-Function and associate it to a formula such as and a wave-form such as . The first perspective is the arithmetic perspective from which things are distributed, unit by unit, in mathematical and physical space and in time (the time perspective), the second perspective (the frequency perspective) is a “trans-arithmetic” perspective from which everything is in some sense always present. From the fact that the initial, foundational, state of the universe is state such that there is no space and time and no separation between abstract and/or physical units (a state such that prime and energy-density are infinite), it follows that the second perspective has priority over the first, that the first perspective is a well-founded illusion -a projection- arising from the second and governed by the Riemann Hypothesis.

The 2-Slit Experiment

In the fundamental experiment of quantum mechanics -the 2-slit experiment- particles are shot particle by particle through two slits and their arrival is registered on a screen. If we look -if a detector shines light on particles as they pass through the slits- we get a pattern indicating that light can be broken down into discrete element, and that these elements pass through the slits like bullets passing through a pair of windows; but if we don’t look, we get a pattern indicating that light is a continuous entity, and that it passes through the slits like water passing through a pair of sluice gates. This second kind of a pattern is known as an “interference pattern” and this is a series of dark/light bands: when the peak of one wave interferes with the trough of another they cancel (destructive interference) resulting in a dark band on the screen; and when the peaks of two waves interfere with each other they reinforce (constructive interference) resulting in a light band on the screen.

The great error of quantum mechanics, and the main reason people have struggled to make sense of it, is the assumption that the first pattern is not an interference pattern. This assumption is false. Contrary to widespread opinion, the first pattern is an inference pattern, but an interference pattern of the RH-friendly, symmetric type. To see what I mean, consider firstly what it is to be balanced (symmetric, classical), and what it is to depart from balance (asymmetric, quantum) by re-expressing the tradition equation for a circle of area 1 () as

Where the traditional equation fails by implying that an energy source located at the center of this area unit-circle is undiminished from center to circumference (it has either a zero or an infinite radius), the second provides us with a potentially infinite hierarchy of energy levels that are necessarily non-infinite and non-zero. Given that gamma is a spacial case of for s =1, we can go from to the more general

If and only if s = 1, the unit above can be subject to an unlimited number of repetitions. If s is a real number greater or less than 1, then this number of repetitions is strictly finite, and the superpositions associated to these strictly finite progressions are asymmetric:

Consider secondly that all continuous arithmetic progressions, and all the continuously countable phenomena corresponding to them, can be described as arising from superpositions such as

So also can random phenomena such as the detected particles in a 2-slit experiment – one perspective is the familiar, arithmetic, perspective, and the other is the trans-arithmetic perspective as this concerns symmetric superpositions. When counting particles, we assume the uniformity and continuity of an arithmetic progression, and note that there is an associated error of square-root size. But this assumption is unwarranted. If instead we assume that there is perfect balance of slit-1 and slit-2 particles, the square root-sized error is transferred to the uniformity of the arithmetic progression. If the error gets any bigger or smaller than this, then the progression loses its continuity. At root of the conservation of this error are the facts that if we assume the uniformity and continuity of an arithmetic progression to count primes, there is an associated error of square-root size, and that although this error is apparently reduced by the addition of harmonic and logarithmic functions to a prime-finding formula such as

it is during the course of these additions really being transferred to the arithmetic progression, and particularly to uniformity of this progression. This error is an ineliminable element of arithmetic consciousness because, as we have seen, arithmetic consciousness emerges from it. But for the error, there is no arithmetic consciousness, and if and when the error becomes smaller of greater, arithmetic consciousness disappears. This is because, beneath these continuous arithmetic progressions (s=1) there are discontinuous ones (s!=1), i.e. beneath the underlying symmetric superstitions (s=1), there are asymmetric superpositions (s!=1) familiar from the pattern formed by the undetected particles in the 2-slit experiment. It has many more champions than not, but scalable QC must have always seemed to minds wary of any proposal to build a tower that reaches to heaven, a highly dubious affair, like perpetual motion, and speed-of-light travel, too good to be true in a world governed by the arrow of time. And yet there is nothing in quantum mechanical theory as it is taught at the time of writing that forbids it, and so anyone who believes that QM is a more or less satisfactory theory would be rationally committed to this belief that scalable qauntum computers can be built. But this is not a proof that QC is possible. Rather, it is a proof that QM is deeply flawed, a reductio ad absurdum of QM…. The big problem -the insurmountable problem- for the billion dollar industry is simply that scalable means arithmetically continuous, arithmetically continuous means asymmetric in the sense of the arrow of time leading from prime density to sparsity symmetric (the arithmetic perspective) and symmetric in the sense of the underlying superposition (the trans-arithmetic perspective), and arithmetically symmetric + trans-arithmetically symmetric means classical, i.e. “scalable” and “quantum” are incompatible terms, and “scalable qauntum computing” is a contradiction in terms. We have a very nice analogy at out disposal… There is a well-know limitation on the maximum share of the computing power of the bitcoin network that any one set of computers in the network can possess before that it ceases to be decentralized. If any one group of users were to control more than 50% of the network’s mining hash-rate, then the possibility exists that they could set up a financial dictatorship, thereby destroying the feature of cryptocurrencies that make them so attractive to those that oppose the inequitable distribution of wealth and power – decentralization.

Here it has been argued that there is a similar decentralization-born limitation on the building scalable quantum computers, that this arises from the limitation there is on the maximum share of the computing power of the “quantum network” underlying all human experience that any that can be possessed by any one set of classical devices, a share limited by the value

and by the Riemann Hypothesis. Public key cryptography, it follows, does the job it was devised to do. Kudos to Rivest, Shamir and Adleman who used Fermat’s Little theorem for something far more useful than anything his “big” theorem has heretofore been used for (16), and to Satoshi Nakamoto who made public key cryptography into something far more than a clever way to stop bad people from stealing stuff (17).

(11) Riemann, G (1859), On the Number of Primes Less Than a Given Magnitude

(12) Dirichlet, P (1837), Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers

Abstract According to the ancient form of Jewish mysticism known as “Kabbalah”, the finite human world is an emanation of an infinite divine realm sometimes known as “Ein Sof” (1). Here a mathematical-physical framework in virtue of which Kabbalistic metaphysics makes perfect sense is outlined.

We begin with Euler’s classic argument (2) that the product continued to infinity of this fraction

in which the numerators are prime numbers and the denominators are one less than the numerators, equals the sum of the infinite series

and they are both infinite. To prove his point to Euler invites us to imagine the extraction from the second series a prime denominator and all remaining multiples of that prime denominator until everything except the first term 1 has been eliminated. Let

Then

This leaves

To eliminate the denominators that are divisible by 3, we divide both sides to get

Subtracting again eliminates all remaining denominators that are multiples of 3 leaving

Applying this eliminatory process to all of the prime numbers leaves

This is a thought-experiment -mere imagination- but if these eliminations could be performed in the physical world, they would result in the disappearance of any distinction between the form and the content of a coordinate system, and therefore the shrinking of space and the slowing of time to a zero-dimensional point. With all of reality contracted to a zero-dimensional point, the distinction between the world and the mind that surveys it is lost. This is the singularity at root of general relativity and Kabbalah. It is not -as is often maintained- all of the mass in the universe compressed down to a point, an infinitely heavy object. This is a gross misconception resulting from the assumption of atomism, and the prioritization of space over light, a misconception responsible for all that is stupid and ugly in contemporary physics, including black holes as infinite at their centers (3), the incongruity of relativity and quantum mechanics (4), the infinities of quantum field theory (5), and the flat rotation curves of distant galaxies that seem to call for dark matter (6, 7)… to name a handful of significantly troublesome things. Rather, this singular state is a state such that all of the space and time -all the gaps and holes- in the universe are from the perspective of this singularity excised, leaving only infinitely concentrated light ( Ohr Ein Sof), an infinitely light object in both senses of the term “light”. On this account there are two forms of curvature resulting either from an imbalance of light and space in favor of light (the most extreme form of this imbalance is the origin of the universe) or from an imbalance of light and space in favour of space (the most extreme forms of this imbalance are black holes).

The first idea we can take from Euler’s thought-experiment is that, since both prime-density and energy-density must at this point be infinite, the spatio-temporal development of the universe from a central singular point towards an ever-increasing state of decentralization is a process involving the distribution of the prime numbers. We can add to this that this process of decentralization takes place according to something known as the Riemann Hypothesis (8, 9), which says that the thinning of primes -the spreading of prime-energy over time and space- with arithmetic increase cannot exceed the upper and lower bounds such as those marked in red and blue in the graphs below:

The Generalized Riemann Hypothesis (10, 11) extends the Riemann Hypothesis by reference arithmetic progressions associated with the equation q n + a where q and n have no common factor greater than 1. In a universe whose fundamental condition is an infinite state of prime and energy density which is diffused from the point of view of any and every frame of reference according to the Generalized Riemann Hypothesis, time has a forwards direction associated with the loss of prime and energy density and a backwards direction associated with a gain in prime and energy density. Because the loss of prime-density predominates over any gains in prime-density, the direction of time is given by the GRH, a balance of the two forms of curvature is maintained by the equality

The waveform () we obtain by taking difference between the number of primes not greater than x and an approximating formula such as

and dividing by x is a superposition of smooth waves:

Or from another perspective on the same phenomenon, we can obtain the superposition by taking the difference between the primes themselves and an approximating formula such as

The intimate connection there is between number theory and quantum field theory can be simply illustrated (12) by associating the creation operators and to the prime numbers … Now we have identified the unique ‘factorization’ of a state into creation operators acting on the ‘vacuum’ with the unique factorization of an integer into prime numbers (and we have a hierarchy of states: |1> is the ‘vacuum’; |2> and |3> and |5> are one-particle states; |6> is a two-particle state… and so on). By reference to the Witten index -the number of bosonic minus the number of fermionic zero-energy states- we see that the Mobius inversion function

={1 = n has an even number of distinct factors, -1 = n has an odd number of distinct factors, 0 = n has a repeated factor}

is equivalent to the operator that distinguishes bosonic from fermionic states, with = 0 when n has a repeated factor being equivalent to the Pauli exclusion principle. If we re-express the Mertens function (which sums the 1s and -1s of the Mobius function) as , we see that sums of these states give us essentially the same composite spiral-wave as before.

Assuming that there are an equal number of non-zero-energy bosonic and fermionic states, this wave depicts the zero-energy fluctuations of these particles, the energy fluctuations of the vacuum. This is to say that the vacuum is the basis of everything -everything emanates from the vacuum- and that the vacuum is far from vacuous. The intersections of the x-axis by these fluctuations can be identified with the collapse of asymmetrical wave-functions… These examples concern the case where q =1 and a = 0, but we can easily construct approximating formulas and wave-forms for all the other possible values of q and a. The value , associated as it is with a very particular balance of prime-density and sparsity, signifies the dividing line between classical objects that are constrained to travel in time in an arithmetic manner, and quantum objects that transcend arithmetic and are not so constrained. A key aspect of this mathematical scheme is that we can take any arithmetic progression associated to q n + a and to an L-Function and associate it to a formula such as and a wave-form such as . The first perspective is the arithmetic perspective from which things are distributed, unit by unit, in mathematical and physical space and in time (the time perspective), the second perspective (the frequency perspective) is a “trans-arithmetic” perspective from which everything is in some sense always present. From the fact that the initial, foundational, state of the universe (Ohr Ein Sof) is state such that there is no space and time and no separation between abstract and/or physical units (a state such that prime and energy-density are infinite), it follows that the second perspective has priority over the first, that the first perspective is a well-founded illusion -a projection- arising from the second and governed by the Riemann Hypothesis. The first perspective -the time perspective- is man’s perspective, but the second perspective -the frequency perspective- is G-d’s perspective.

(10) Dirichlet, P (1837), Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers

Abstract Some connections there are between number theory -especially the Riemann Hypothesis- and physics are explored.

We begin with Euler’s classic argument (1) that the product continued to infinity of this fraction

in which the numerators are prime numbers and the denominators are one less than the numerators, equals the sum of the infinite series

and they are both infinite. To prove his point to Euler invites us to imagine the extraction from the second series a prime denominator and all remaining multiples of that prime denominator until everything except the first term 1 has been eliminated. Let

Then

This leaves

To eliminate the denominators that are divisible by 3, we divide both sides to get

Subtracting again eliminates all remaining denominators that are multiples of 3 leaving

Applying this eliminatory process to all of the prime numbers leaves

This is a thought-experiment -mere imagination- but if these eliminations could be performed in the physical world, they would result in the disappearance of any distinction between the form and the content of a coordinate system, and therefore the shrinking of space and the slowing of time to a zero-dimensional point. With all of reality contracted to a zero-dimensional point, the distinction between the world and the mind that surveys it is lost. This is the singularity at root of general relativity. It is not -as is often maintained- all of the mass in the universe compressed down to a point, an infinitely heavy object. This is a gross misconception resulting from the assumption of atomism, and the prioritization of space over light, a misconception responsible for all that is ugly in contemporary physics, including black holes as infinite at their centers (2), the incongruity of relativity and quantum mechanics (3), the infinities of quantum field theory (4), and the flat rotation curves of distant galaxies that seem to call for dark matter (5, 6)… to name a handful of significantly troublesome things. Rather, this singular state is a state such that all of the space and time -all the gaps and holes- in the universe are from the perspective of this singularity excised, leaving only infinitely concentrated light, an infinitely light object in both senses of the term “light”. On this account there are two forms of curvature resulting either from an imbalance of light and space in favor of light (the most extreme form of this imbalance is the origin of the universe) or from an imbalance of light and space in favour of space (the most extreme forms of this imbalance are black holes).

The first idea we can take from Euler’s thought-experiment is that, since both prime-density and energy-density must at this point be infinite, the spatio-temporal development of the universe from a central singular point towards an ever-increasing state of decentralization is a process involving the distribution of the prime numbers. We can add to this that this process of decentralization takes place according to something known as the Riemann Hypothesis (7, 8), which says that the thinning of primes -the spreading of prime-energy over time and space- with arithmetic increase cannot exceed the upper and lower bounds such as those marked in red and blue in the graphs below:

The Generalized Riemann Hypothesis (9, 10) extends the Riemann Hypothesis by reference arithmetic progressions associated with the equation q n + a where q and n have no common factor greater than 1. In a universe whose fundamental condition is an infinite state of prime and energy density which is diffused from the point of view of any and every frame of reference according to the Generalized Riemann Hypothesis, time has a forwards direction associated with the loss of prime and energy density and a backwards direction associated with a gain in prime and energy density. Because the loss of prime-density predominates over any gains in prime-density, the direction of time is given by the GRH, a balance of the two forms of curvature is maintained by the equality

The waveform () we obtain by taking difference between the number of primes not greater than x and an approximating formula such as

and dividing by x is a superposition of smooth waves:

Or from another perspective on the same phenomenon, we can obtain the superposition by taking the difference between the primes themselves and an approximating formula such as

The intimate connection there is between number theory and quantum field theory can be simply illustrated (11) by associating the creation operators and to the prime numbers … Now we have identified the unique ‘factorization’ of a state into creation operators acting on the ‘vacuum’ with the unique factorization of an integer into prime numbers (and we have a hierarchy of states: |1> is the ‘vacuum’; |2> and |3> and |5> are one-particle states; |6> is a two-particle state… and so on). By reference to the Witten index (12) -the number of bosonic minus the number of fermionic zero-energy states- we see that the Mobius inversion function

={1 = n has an even number of distinct factors, -1 = n has an odd number of distinct factors, 0 = n has a repeated factor}

is equivalent to the operator that distinguishes bosonic from fermionic states, with = 0 when n has a repeated factor being equivalent to the Pauli exclusion principle. If we re-express the Mertens function (which sums the 1s and -1s of the Mobius function) as , we see that sums of these states give us essentially the same composite spiral-wave as before.

Assuming that there are an equal number of non-zero-energy bosonic and fermionic states, this wave depicts the zero-energy fluctuations of these particles, the energy fluctuations of the vacuum. This is to say that the vacuum is the basis of everything -everything emanates from the vacuum- and that the vacuum is far from vacuous. There is here a distinction between symmetric superpositions such as (where the associated progression is arithmetically continuous) and asymmetric superpositions familiar from quantum mechanics (where the associated progression is arithmetically discontinuous). The intersections of the x-axis by these fluctuations can be identified with the end of one asymmetric superposition and the beginning of another… These examples concern the case where q =1 and a = 0, but we can easily construct approximating formulas and wave-forms for all the other possible values of q and a. The value , associated as it is with a very particular balance of prime-density and sparsity, signifies the dividing line between classical objects that are constrained to travel in time in an arithmetic manner, and quantum objects that transcend arithmetic and are not so constrained. A key aspect of this mathematical scheme is that we can take any arithmetic progression associated to q n + a and to an L-Function and associate it to a formula such as and a wave-form such as . The first perspective is the arithmetic perspective from which things are distributed, unit by unit, in mathematical and physical space and in time (the time perspective), the second perspective (the frequency perspective) is a “trans-arithmetic” perspective from which everything is in some sense always present. From the fact that the initial, foundational, state of the universe is state such that there is no space and time and no separation between abstract and/or physical units (a state such that prime and energy-density are infinite), it follows that the second perspective has priority over the first, that the first perspective is a well-founded illusion -a projection- arising from the second and governed by the Riemann Hypothesis.

(9) Dirichlet, P (1837), Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers

Abstract An ancient dilemma for Christianity is that a) the Jewish Messiah must according to prophecy be a member of the house of David, and the tribe of Judah, but b) house and tribal identity are inherited only through the male line, and Jesus lacked a human father, albeit that Joseph belonged to the house of David. Here we offer a resolution of this dilemma, a resolution that arises in the context of a non-Darwinian approach to genetic genealogy.

PART 1

The Conflict Between the Davidic Descent of Messiah and the Virgin Birth of Jesus

One issue which creates great difficulties for those that maintain Jesus of Nazareth is the Jewish Messiah (“Mashiach”) is that Messiah, by prophecy, and by tradition, must belong to the house of David.

A shoot will come up from the stump of Jesse; from his roots a Branch will bear fruit.

Since Jesus was born of a virgin, since house and tribal identity are inherited only through the father’s line, how can it be that Jesus fulfills the prophecy that Messiah is from the house of David? It may seem that the only answers Christians and Messianic Jews have at their disposal are that 1) Jesus was the legal, but not the genetic descendant of David, through Joseph and/or that 2) Mary was a genetic descendant of the house of David.

There are two inconsistent genealogies given for Jesus, one in Mathew 1: 6, and another in Luke 3: 31. Judging by the text, both genealogies trace the descent from David through Joseph, but it may seem that there is a need for Christians and Messianics to claim that one or other of these genealogies is through Mary. Jews for Jesus:

Unlike Matthew, Luke follows strict Jewish procedure and custom in that he omits no names and mentions no women. However, if by Jewish custom one could not mention the name of a woman, but wished to trace her line, how would one do so? He would use the name of her husband. (Possible Old Testament precedents for this practice are Ezra 2:61 and Nehemiah 7:63.) That would raise a second question: If someone studied a genealogy, how would he know whether the genealogy were that of the husband or that of the wife, since in either case the husband’s name would be used? The answer is not difficult; the problem lies with the English language.

In English it is not good grammar to use a definite article (“the”) before a proper name (“the” Matthew, “the” Luke, “the” Miriam): however, it is quite permissible in Greek grammar. In the Greek text of Luke’s genealogy, every single name mentioned has the Greek definite article “the” with one exception: the name of Joseph (Luke 3:23). Someone reading the original would understand by the missing definite article from Joseph’s name that this was not really Joseph’s genealogy, but his wife Miriam’s.

To circumvent the objection that Mary’s genealogy is irrelevant because house and tribal are identity inherited only through the father’s line, Jews for Jesus argue that

Whereas Jewish nationality and tribal identity were normally determined by the father, with the Messiah it would be different. Since he was to have no human father, his nationality and his tribal identity would come entirely from his mother. True, this is contrary to the norm, but so is a virgin birth. With the Messiah, things would be different.

Maybe it is true that the genealogy in Luke is Mary’s, there is a dearth of biblical evidence to this effect. In the first place, Luke explicitly says that the genealogy is Joseph’s, and gives no indication that it is Mary’s. Luke 3: 23:

When He began His ministry, Jesus Himself was about thirty years of age, being, as was supposed, the son of Joseph, the son of Eli, the son of…

And elsewhere he stresses that it is Joseph -not Mary- that is the descendant of David. Luke 1: 27:

… a virgin engaged to a man whose name was Joseph, of the descendants of David; and the virgin’s name was Mary.

Luke 2: 4-5:

Joseph also went up from Galilee, from the city of Nazareth, to Judea, to the city of David which is called Bethlehem, because he was of the house and family of David, in order to register along with Mary, who was engaged to him, and was with child.

As Messianic Jew, Rabbi Baruch Korman at LoveIsrael.Org says, it is “really dangerous” to say that the genealogy given in Luke is Mary’s:

You know what’s really dangerous about that? Is that Mary doesn’t even appear in it. And its very dangerous to say “Oh, This is Mary’s lineage, when we have no biblical evidence. Do you see anything in the text that would support that? It is simply a supposition that has no scriptural foundation.

Moreover, we know from the example of John the Baptist’s mother Elizabeth (who is identified in Luke 1: 5 as a descendant of Aaron) that, if it was true and relevant that Mary was a descant of David, wouldn’t Luke simply have said that she was? There is not a shred of biblical evidence to the effect that Luke’s genealogy is intended to be Mary’s rather than Joseph’s, and the example of Elizabeth suggests that the reason Luke did not say that Mary was a descendant of David is that this is either untrue or irrelevant. Jews For Jesus popular argument looks like a case of the old adage that, if you torture the data hard enough, it will confess to anything you want, and the truth is that Christians/Messianics arguing in this way are trying to make are trying to harmonize the idea of Davidic descent of Jesus and the virgin birth by revising other deeper beliefs in their belief-structure in a way that is logically awkward, has little to no biblical support, and makes little to no sense in the context of Judaism.

The Curse of Jeconiah

One reason for these inconsistent genealogies may be the “The Curse if Jeconiah”. King Jeconiah, or Jehoiachin, or Coniah, was a King of Judah who was cursed by God. Jeremiah 22:

Record this man as if childless, a man who will not prosper in his lifetime, for none of his offspring will prosper, none will sit on the throne of David or rule anymore in Judah.

Since Joesph’s connection to David was, according to Mathew, through Jechoniah it follows that if Jesus inherits his Davidic status through Joseph, he is not allegible to be Messiah, who must sit on this throne forever. A commonly proposed solution is to appeal to fact that Luke’s genealogy doesn’t go through Solomon and Jechoniah, but through another son of David -Nathan- and to attribute this genealogy to Mary, and argue that Jesus has no blood connection to Jechoniah. But one might just as easily attribute Mathew’s genealogy to Mary, and Luke’s genealogy to Joseph. Clement of Alexandria did this, writing (1):

And in the Gospel according to Matthew, the genealogy which begins with Abraham is continued down to Mary the mother of the Lord.

As did Victorinus of Pettau (2):

Matthew strives to declare to us the genealogy of Mary, from whom Christ took flesh.

Rabbi Baruch’s approach is to say that Luke’s genealogy is -as it appears- a genealogy of Joseph, not Mary, and that Luke provides an alternative to Mathew’s potentially illegitimate genealogy by going through Nathan rather than Solomon and thus guarding against the accusation that the descent from Jeconiah is disqualifying:

Now is it a problem that Joseph is the son of, what it says here, Eli? No. Because we’re going to find out something. This gospel, when it speaks about the genealogy, it makes a very important change. What is that? Well here again, one of the driving forces, not only in Mathews’s genealogy, but in Luke’s genealogy, is this problem with Jehoiachin, that evil king, the one that cannot have any of his heirs siting upon the throne of David. So what we’re going to see is this. We’re going to see that in Luke’s gospel there’s a change. When we come to David, the next generation in Mathew’s gospel, who is the person mentioned? King Solomon. Luke says “No – I am not taking the genealogy through Solomon. I’m going to take it through another son of David.” And that is who? Nathan. So everything we have… from the son of David, the generation after David, up until now, with the exception of Joseph is different. Now why is that important? Because when you look at all those verses… none of those things agree with anything in the old testament. That’s OK. Because we don’t have Nathan’s genealogy. Luke knows it. He got it from the temple, but there’s nowhere else it’s mentioned. Now why is that important to make such a statement? Because all of these attacks for differences between Mathew and Luke’s gospel in regard to the genealogy, when you say it’s a different lineage, there would not be any congruence. It’s going to be different.

Rabbi Baruch’s overall approach is that biblical genealogies are not necessarily literalistic. Some genealogies in both the old and the new testaments are there, he believes, to create informative pictures for us, rather than to precisely detail the members of lineages. We must, he believes, satisfy ourselves with the idea that, although Jesus was not a genetic descendant of David, he was -in the same way that an adoptee is the legal decadent of their adopted parents- a legal descendant. No doubt it is true of some genealogies in the bible that their purpose is illustrate truths, but is this really what the prophet meant when he said that “A shoot will come up from the stump of Jesse; from his roots a Branch will bear fruit.”? Maybe, but I doubt it. And there is a further problem. 1 Chronicles (22: 8 – 10) tells us that

But this word of the Lord came to me: ‘You have shed much blood and have fought many wars. You are not to build a house for my Name, because you have shed much blood on the earth in my sight. But you will have a son who will be a man of peace and rest, and I will give him rest from all his enemies on every side. His name will be Solomon, and I will grant Israel peace and quiet during his reign. He is the one who will build a house for my Name. He will be my son, and I will be his father. And I will establish the throne of his kingdom over Israel forever.

The genealogy in Mathew passes through Solomon and therefore relates to the covenant, but it also passes through Jeconiah. The genealogy in Luke, which passes through David’s son Nathan rather than Solomon, and therefore bypasses Jeconiah, doesn’t relate to the covenant.

The Genetic Solution

The dilemma faced by Christians and Messianics may look on the face of things to be unsolvable. It may look as if they have no choice but to fudge their way out of the dilemma, by allowing either that the descent of Jesus from David is legal, but not genetic, and/or in so far as it is genetic it is not through the father’s line… But there is yet at one further argument, one last stand if you will, that has not been considered, and which solves all dilemmas without fudging.

Looking closely at the Isaiah’s text, we see that the prophecy doesn’t say that Messiah will be a literal descendant of King David – it says only that Messiah will be part of the “tree” whose “root” is King David’s father Jesse:

A shoot will come up from the stump of Jesse; from his roots a Branch will bear fruit.

This is a genealogical tree of course, but it is also a genetic tree. Normally these trees correspond to each other, so that for each branch on the one tree there is a corresponding branch on the other, but in the case someone who enters the world from outside in a direct manner, the genealogical tree may be disrupted. Let me explain with reference, firstly to blood samples taken from the Turin Shroud during the 1978 STURP investigation by micro-analyst Professor Giovanni Riggi , who used scalpels to take blood samples from the lower part of the apparent crown of thorns bloodstains on the Shroud’s dorsal image. Ian Wilson quoted from a June/July 1996 editorial for the British Society For The Shroud of Turin:

…The story then moves on four and a half years, to Garza-Valdes’s discovery of a partly living bioplastic coating on certain ancient Mayan artefacts that he found to have caused some serious radiocarbon dating errors. This caused him to wonder whether the Shroud might have become covered with a similar accretion that had led to similar errors, and on his journeying to Turin expressly to gain permission to examine the Shroud directly, he was initially turned away by Cardinal Saldarini, Ballestrero’s successor as the Shroud’s custodian. However Professor Gonella, who had been Cardinal Ballestrero’s foremost scientific advisor, directed Valdes to Giovanni Riggi, who duly brought his samples out from the bank vault. On examining these under a portable microscope Valdes immediately saw that there was indeed a bioplastic coating quite sufficient to have seriously affected the accuracy of the Shroud carbondating reading. He then offered Riggi the services of the University of Texas’s Health Science Center at San Antonio, Texas, for further studies, in particular of Riggi’s samples from the Shroud’s crown of thorns bloodflows. And with little further ado Riggi was westwardbound across the Atlantic, the samples in his luggage.

Once arrived at the University of Texas’s San Antonio campus, Riggi’s ‘blood’ samples swiftly came under the scrutiny of two individuals new to Shroud circles, the University’s microbiology professor Stephen J.Mattingly, widely respected for his research on group B streptococci and neonatal disease, and his assistant professor, Dr.Victor V.Tryon, director of the University’s Center for Advanced DNA Technologies. Inevitably one of the first questions to be addressed was whether that which appeared to be blood on the samples was indeed actual blood, and whether it derived from a human being? Studying a sticky tape bearing a 1.5mm ‘blood’ fragment Dr.Tryon unhesitatingly confirmed it as human blood, carrying both the X and Y chromosomes that indicate male sex. A second fragment furnished an identical result…

Later the samples were cloned and deposited in the University of Texas’s blood bank, and as Ian Wilson puts it, “three American scientists are holding a genetic code that they believe once controlled every cell in Jesus’s physical body, details of which they can open up at any time…” (3) Whatever the origin of his Y-chromosome, there can be no question that Jesus possessed human Y-DNA. All human Y-DNA can be classified according to its place on a genetic haplogroup tree, tracing back to Y-Adam, from which it follows that Jesus’ Y-DNA must have a location on a tree such as that below:

There is no theoretical problem with the idea of injecting an egg with Y-DNA, thereby producing a clone of sorts, which means we can simply assert that Jesus’ Y-DNA was Davidic, but transmitted directly to Mary, rather than through Joseph, solving the first dilemma: as required by the Isaiah prophecy, Jesus is a “branch” on the patrilineal genetic tree that extends back to the “root” of Jesse and so he is entitled on this basis to be Messiah. By contrast with the prophecy in Isaiah, which refers only to a tree which has Jesse at its root, not to descendants, the curse of Jeconiah recorded in Chronicles refers specifically to the descendants of Jeconiah:

Record this man as if childless, a man who will not prosper in his lifetime, for one of his offspring will prosper, none will sit on the throne of David or rule anymore in Judah.

The first connection (between Jesse and Messiah) can from Isaiah’s text be genetic without being genealogical, while the second connection (between Jeocniah and his descendants) is clearly genealogical as well as a genetic. If Jesus’ Davidic Y-DNA was transmitted directly rather than indirectly to Mary (the virgin birth), this means that Jesus does not appear on the genealogical tree tracing Jeconiah and his descendants, and is therefore a member of the house of David and Solomon, but not subject to the curse of Jeconiah. By the same logic, Jesus does not appear on the genealogical tree tracing David and his descendants, but Isaiah’s text requires only that Messiah be from the tree at whose root is Jesse, a tree which can be purely genetic in character. So the second dilemma is solved also. As to the third dilemma, the passage in Chronicles doesn’t say that a genealogical descendant of David through Solomon will sit on the throne of Israel forever, it says only that this throne will be established forever:

… His name will be Solomon, and I will grant Israel peace and quiet during his reign. He is the one who will build a house for my Name. He will be my son, and I will be his father. And I will establish the throne of his kingdom over Israel forever.

This means that a man possessing Davidic/Solomonic Y-DNA who sits on the throne of Israel forever fully satisfies the description in Chronicles, and so this dilemma also is solved. Suppose for the sake of illustration that a complete genetic clone of King David is created. If the David-clone sits on the throne of Israel forever, is it not true that the throne of David is thereby established forever? Similarly, is it not true both that the David-clone is a genuine member of the house of David, and that he is not a descendant of Jeconiah? That Messiah is a part David-clone is suggested by Ezekiel 37:24:

My servant David will be king over them, and they will all have one shepherd. They will follow my laws and be careful to keep my decrees.

No one is a better candidate for the role of David-clone than the man who, given the reality of the virgin birth, must have been a clone. From man’s point of view, this scheme of things is something that could not have been understood any earlier than the 20th century, when DNA was discovered and the science of genetics founded…

Abstract The Torah makes no direct reference to it, yet there is evidence -drawn from the Jewish and Christian bibles, and from other sources- that reincarnation has been a traditional aspect of Judaism for a long time…

The Kabbalah and Gilgul

Kabbalah is a set of doctrines, of Jewish origin, concerning the relationship between the infinite realm of God (Ein Sof (סוף)), and God’s creation, the finite realm of man (1). According to Ramak, a central figure in the development of Kabbalistic Judaism, souls cycle through lives or incarnations, being attached to different human bodies over time, depending particularly on the soul’s task in the physical world, and the spiritual levels of the bodies previously occupied by the soul as it journeys through the stations depicted in the “tree of life” below (2):

The word Jews use for reincarnation is “gilgul”, which means in Hebrew means “cycle” or “wheel”, appears neither in the written nor in the oral Torah, leading some rabbis to maintain that reincarnation has no place in Judaism. Kabbalists by contrast maintain that gilgul and reincarnation have always been an important part of Judaism, but a hidden or secret part.

Evidence of the Jewish Belief in Reincarnation in the Bible

Whilst it is true that there is no mention of gigul in the Torah, there are passages in the bible -old testament and new- that make no sense unless reincarnation was indeed an intimate part of Jewish tradition. Malachi 4: 5:

See, I will send the prophet Elijah to you before that great and dreadful day of the LORD comes.

This reads as if the returning Elijah is the literal Elijah, not merely someone with a similar personality or mission. Knowing that the return of Elijah precedes the coming of Messiah, the disciples ask Jesus in Mathew 17:10:

Why then do the teachers of the law say that Elijah must come first?

Jesus reply (Mathew 17:11-13) is that

To be sure, Elijah comes and will restore all things. But I tell you, Elijah has already come, and they did not recognize him, but have done to him everything they wished. In the same way the Son of Man is going to suffer at their hands.

Then the disciples understood that he was talking to them about John the Baptist. In what sense was John the Baptist the same as Elijah? Was he the reincarnation of Elijah? Luke 1: 17 says:

And he will go on before the Lord, in the spirit and power of Elijah, to turn the hearts of the parents to their children and the disobedient to the wisdom of the righteous – to make ready a people prepared for the Lord.

This refers back to Malachi, since Malachi 4: 6 says of a future Elijah that

He will turn the hearts of the parents to their children…

One who comes in the spirit and power of another might well be the reincarnation of that other person -a person comes in the spirit and the power of themselves- and John 1: 6 says that

There was a man sent from God whose name was John.

which suggests that John pre-existed his arrival on earth. But John 1: 21 has John the Baptists himself explicitly deny being Elijah. :

Are you Elijah?” He said, “I am not.” “Are you the Prophet?” He answered, “No.

Jewish historian Josephus writes of the Pharisees (3) that they

say that all souls are incorruptible; but that the souls of good men are only removed into other bodies; — but that the souls of bad men are subject to eternal punishment.

Thackeray (4) says that this is an expression of “the doctrine of the reincarnation of the soul”, and although it has been argued that is Josephus talking about reincarnation in the sense of the souls being reincarnated in another earthy as opposed to a heavenly body, his account agrees with that of the Safed Kabbalists, Ramak (Moshe Cordovero) and Arizal (Isaac Luria) (5), according to whom reincarnation is a privilege for those who have made an effort to be good in their lives (some sinners, according to Ramak and Arizal, are never reincarnated). This passage in John, together with another passage in which Jesus asks his disciples who people say he is and is told “Some say John the Baptist; others say Elijah; and still others, and Jeremiah or one of the prophets.” (Mathew 16: 13-14), doesn’t make sense unless the Jews of Jesus’ time possessed a belief in reincarnation in this sense of an earthy reincarnation. Jesus’ identity as John the Baptist might have been a matter of resurrection (At that time Herod the Tetrarch heard the reports about Jesus, and he said to his attendants, “This is John the Baptist; he has risen from the dead! That is why miraculous powers are at work in him.” (Mathew 14: 1 – 2)), but the only possible way Jesus could have been Elijah, Jeremiah, or one of the other prophets referred to is if he was a reincarnation of these men. We can conclude then that, although John in fact denied being Elijah, he might just as easily have affirmed it, i.e. his denial was not because there could be no such thing as a reincarnated Elijah but because he didn’t believe -or pretended he didn’t believe- he was that person. I say “pretended” because while, on the one hand, John denied being Elijah, on the other hand he identified himself with Elijah by quoting Isaiah 40: 3:

I am the voice of one calling: “In the wilderness prepare the way for the LORD; make straight in the desert a highway for our God.

This preparation of the way is very the role of the returned Elijah spoken of in Malachi. Malachi 3: 1:

I will send my messenger, who will prepare the way before me…

His reply is contradictory: “I am not Elijah, yet I am the voice of one calling “In the wilderness prepare the way for the LORD; make straight in the desert a highway for our God, i.e. Elijah.””

The Science of Reincarnation

The question of whether reincarnation is -as Hasidic Jews insist- part of Judaic tradition and the question of whether reincarnation is a reality are independent, and a way to approach the latter question is through the scientific evidence that has been accumulated for reincarnation. A well known possible example of reincarnation that of Shanti Devi born in Dehli India in 1926. She never spoke until the age of four years old at which time she told her parents that her real home was in Mathura where her husband lived, 140 km away (6). She said was named “Lugdi” Devi and married to a merchant named “Kedar Nath” and that she died ten days after having given birth to a child, details which proved to be accurate. When Mahatma Gandhi heard about the case, he met Shanti Devi and set up a commission to investigate (7). The commission traveled with Shanti Devi to Mathura where she recognized several family members, including the grandfather of Lugdi Devi and decided that she was indeed the reincarnation of Lugdi Devi. Numerous examples of the same phenomenon come from the research begun in 1960 by the late chairman of the department of psychiatry at the University of Virginia Dr Ian Stephenson (8). Stephenson’s overall thesis is that was that children’s possession of certain knowledge together with unusual abilities, philias, phobias, certain physical defects and illnesses, cannot not be fully explained by heredity or the environment, and that reincarnation provides a third and the best type of explanation (9). His research has been continued by Dr Jon Tucker, his replacement at the University of Virginia. After studying over 2500 cases of seeming childhood reincarnation, Tucker reports that more than 70% of these cases involve people who died unnatural often violent deaths (10). He also reports that over 35% of children in the unnatural-death cases show an intense fear toward the mode of death of the previous person. A related feature of these cases is the presence on the bodies of some of these children of broth marks corresponding to injuries -often fatal injuries- sustained by the deceased.

We can apply the following list of possibilities above to any seeming case of reincarnation where someone seems to possess knowledge particular to the person they claim to re-instantiate:

Fraud by the reporter of the case, e.g. Stephenson or Tucker.

Fraud by the subject of the case: they guessed the details or they fabricated them based on knowledge and/or research.

The subject somehow has access to a realm in which this information is available, although it isn’t the case that they are reincarnated.

The subject really is reincarnated.

Orthodox Jews and Christians are inclined to insist that the reincarnation is impossible, and that all apparent cases of reincarnation arise because the person supplying the privileged information that seemingly establishes their claim is possessed or influenced by a spirit (which falls under 3). But this theory can’t account for those many cases when -in addition to the possession of privileged information- there are birth marks or deformities on the living person corresponding to the wounds of a dead person or a strong physical resemblance between the living and the dead person.

In 2000, Tucker analyzed 799 cases of children who claim to remember a previous life and found that in the stronger cases there was a greater facial resemblance to the deceased individual that they were said to have been (11).

Interestingly, most of these cases involve an average time of sixteen months between the death of the one person and their seeming reincarnation, and they involve memories that mainly concern events towards the end of the life of the deceased (12). This feature of the cases naturally raises the question of the what happens if and when there is a long time -say centuries- between incarnations, and whether there could be any clear memories of the previous incarnation in these circumstances, i.e the question of whether there is an inversely proportional relationship between the time between incarnations and the strength of the memories, the strength of the abilities and phobias etc., the strength of the birth marks, and the strength of facial resemblances… This is a question that could be researched by use of hypnotic regression using a similar approach as that of Stephenson and Tucker, i.e. gathering the details given by the subject of their past life, and then seeking to disestablish any connection between lives that requires reincarnation to explain it…

Stephenson studied a case in which a small child named Suzanne Ghanem believed she was a woman named Hanan Monsour (8). At sixteenth months, Suzanne grasped the phone as if trying to speak into it and said repeatedly, “Hello, Leila?” When she was older, Suzanne explained that Leila was one of her children in her past life and that she was not Suzanne, but Hanan. By the time she was two, Suzanne had correctly identified her (Hanan’s) other children, her husband, Farouk, and her parents and her brothers from the previous lifetime…

A photographic comparison of the two women made when Suzanne was an adult, and of a similar age to Hanan, reveals a distinct physical resemblance.

An excellent example of the transmission of physical features from one life to another is the facial similarity between Fire Chief Jeffrey Keene and General John B. Gordon. Below we compare Keene with himself and then with Gordon using Microsoft’s facial recognition algorithm (which assigns a number between 0 and 1, where 0 means two faces certainly don’t belong to the same person, and 1 means that they certainly do):

The story of the connection between Keene and Gordon began when Keene walking through a field called “Sunken Road” and was suddenly and unexpectedly overwhelmed by a severe anxiety attack (13):

A wave of grief, sadness and anger washed over me. Without warning, I was suddenly being consumed by sensations. Burning tears ran down my cheeks. It became difficult to breathe. I gasped for air, as I stood transfixed in the old roadbed. To this day I could not tell you how much time transpired, but as these feelings, this emotional overload passed, I found myself exhausted as if I had run a marathon. Crawling up the steep embankment to get out of the road, I turned and looked back. I was a bit shaken to say the least and wondered at what had just taken place. It was difficult getting back to the car because I felt so weak…

A year following this experience, Keene stumbled on a civil war magazine in which there was a picture of Gordon, who was severely wounded in a civil war battle fought on Sunken Road. Thereafter he went on to discover many parallels between himself and Gordon, including a similar physical appearance (looks, height, eye color, birthmarks), personality traits, common lifetime events, writing styles, habits and traits. Keene’s strange experience at Sunken Road by itself is perhaps easily explicable as coincidence, Keene’s close physical resemblance to General Gordon by itself is certainly easily explicable as coincidence, but together they suggest to something other than mere coincidence. Assuming that there is no error or fraud, the possession by people such as Suzanne Ghanem and Jeffrey Keene of privileged information about Hanan Monsour and General Gorden, together with strong physical similarities to these people, imply that reincarnation really has taken place as opposed to an illusion of reincarnation created by a psychic insight into the past.

A compelling case for the reality of reincarnation concerns young boy born in 1998 -James Leininger- who from the age of two claimed to be the reincarnation of an American navy fighter pilot James Huston shot down and killed by the Japanese in 1945 (14).

James (pictured at an older age below left) correctly identified the aircraft carrier from which he which James Huston (below right) was flying his missions and its location at the time Huston was killed, correctly identified one of Huston’s co-pilots, and correctly identified the manner in which Huston was shot down…

The skeptical objection is encapsulated by a comment appearing beneath a you-tube video telling this story:

This is ridiculous. I’ll throw out 3 explanations, what is the most logical:

1. His father is making the whole story up to try and become famous and earn a bit of money. Pretty logical. Everyone likes to make a buck. We didn’t even get to see James speak and be asked questions without his mother and father next to him. That would result in him not being able to answer questions except for the ones mentioned by his father there.

2. He somehow heard about this James Huston guy’s story somewhere. Seeing how he absolutely loved planes, it seems quite possible that some relative could have mentioned this story to him. He wouldn’t have to go in depth, he didn’t reveal that much.

3. When you die, your brain, which holds your memories by the way, somehow gets transfered into the child of a random boy born in the USA.

Now which seems more logical…

This critic is grasping at straws (James’ father is a staunch evangelical Christian, unwaveringly upright, and initially very opposed to the idea of reincarnation, and James Huston’s story only became public knowledge after this case), but the response is typical of the skeptic of reincarnation: regardless of the strength of the evidence, there is something objectionable with the very idea of reincarnation that calls upon the “logical” person to look for alternative explanations.

Reconsider our list of possibilities:

Fraud by the reporter of the case.

Fraud by the subject of the case: they guessed the details or they fabricated them based on knowledge and/or research.

The subject somehow has access to a realm in which this information is available, although it isn’t the case that they are reincarnated.

The subject really is reincarnated.

3 and 4 are out for the you-tube critic and skeptics, leaving them with 1 and 2 and with error and/or fraud. But what exactly is the problem with the other possibilities, and with the possibility that the subject is reincarnated? The answer is simply that the assumption that consciousness is in the brain regarded as a piece of machinery means that when the brain dies there is nothing left of consciousness. Stephenson’s most active critic was for a time was philosopher Paul Edwards, editor-in-chief of Macmillan’s Encyclopedia of Philosophy. From 1986, Edwards punished several articles on Stevenson’s work, and discussed Stevenson in his Reincarnation: A Critical Examination (15, 16). He argued that Stevenson’s views were “absurd nonsense” and that when examined in detail his case studies had “big holes” and “do not even begin to add up to a significant counterweight to the initial presumption against reincarnation.” Stevenson, he claimed, “evidently lives in a cloud-cuckoo-land.” Against Edwards, philosopher Robert Almeder points out in Death and Personal Survival (17) that the philosopher has begged the question by stating in advance that the idea of consciousness existing without the brain in the interval between lives was incredible, and that Edwards’s “dogmatic materialism” has forced him to the view that Stevenson’s case studies must be examples of fraud or delusional thinking… This point gets right to the point: Almeder finds that Edwards doesn’t have anything other than empty rhetoric with which to discredit Stephenson’s research, but if one assumes that consciousness can only exist in a functioning brain then what choice does one have but to dismiss afterlife and reincarnation?

In the well known time-travel paradox “The Grandfather”, a traveler goes back in time and kills his own grandfather, thus removing the grounds of his own existence (it was first proposed by the science fiction writer René Barjavel in 1944 (1)):

Noël Essaillon (physicist-chemist), drawing on the work and collaboration of a young mathematician (Pierre Saint-Menoux), invents a substance (Christmas 3) to travel in time. First developed as capsules to be ingested, it then smeared a suit much better studied for travel time. Saint-Menoux explores first of all the near future then, emboldening himself, a very distant future where he discovers a humanity having evolved towards the exclusive specialization of the tasks. But travel in time is not without danger, and Saint-Menoux will have to learn the hard way that any action has consequences.

The paradox can be generalized by considering that certain past states are conditions of present states, and that altering a past state potentially creates a form of The Grandfather paradox.

There is another time-travel paradox that can be called “The Shakespeare”. In this paradox a traveler goes back in time to give Shakespeare the manuscripts of his plays, thus saving him the trouble of having to write them for himself. The paradox is that the plays have no author: Shakespeare didn’t write them because he was given them by the time-traveler; but the time-traveler didn’t write them either because he was given them by Shakespeare. This paradox is well depicted by Escher’s Drawing Hands:

Imagine going back or forward in time and meeting up with a younger or older self. These are clear-cut cases of what we can call self-intersecting time-lines. The self-intersection can be interpreted as contradictory (The Grandfather) or circular (The Shakespeare) depending on whether or not it is supposed to involve self-interference: if it is supposed to involve self-interference (as it obviously would if one killed one’s ancestor) then the intersection is contradictory; if the self-intersection is supposed not to involve self-interference then it is circular. Corresponding to these two types of self-intersection there are two types of self-intersecting time-lines, one that involves self-interference and one that does not. Corresponding to these two types of self-intersection are two types of particles…

The Riemann Hypothesis, Physics, and Time-Travel

Revisit Euler’s classic argument (10) that the product continued to infinity of this fraction

in which the numerators are prime numbers and the denominators are one less than the numerators, equals the sum of the infinite series

and they are both infinite. To prove his point to Euler invites us to imagine the extraction from the second series a prime denominator and all remaining multiples of that prime denominator until everything except the first term 1 has been eliminated. Let

Then

This leaves

To eliminate the denominators that are divisible by 3, we divide both sides to get

Subtracting again eliminates all remaining denominators that are multiples of 3 leaving

Applying this eliminatory process to all of the prime numbers leaves

This is a thought-experiment -mere imagination- but if these eliminations could be performed in the physical world, they would result in the disappearance of any distinction between the form and the content of a coordinate system, and therefore the shrinking of space and the slowing of time to a zero-dimensional point. With all of reality contracted to a zero-dimensional point, the distinction between the world and the mind that surveys it is lost. This is the singularity at root of general relativity. It is not as is often maintained all of mass in the compressed down to a point, a prioritization of space over light that creates all that is ugly in physics, including black holes as infinite at their centers (3), the incongruity of relativity and quantum mechanics (4), the infinities of quantum field theory (5), and the flat rotation curves of distant galaxies that seem to call for dark matter (6, 7)…

Rather, all of the space and time -all the gaps and holes- in the universe are from the perspective of this singularity excised, leaving only infinitely concentrated light.

The first idea we can take from Euler’s thought-experiment is that, since both prime-density and energy-density must at this point be infinite, the spatio-temporal development of the universe from a central singular point towards an ever-increasing state of de-centralization is a process involving the distribution of the prime numbers. We can add to this that this process of decentralization takes place in terms of something known as the Riemann Hypothesis (8, 9), in terms of which the thinning of primes -the spreading of prime-energy over time and space- with arithmetic increase cannot exceed the upper and lower bounds such as those marked in red and blue in the graphs below:

The Generalized Riemann Hypothesis (10, 11) extends the Riemann Hypothesis by reference arithmetic progressions associated with the equation q n + a where q and n have no common factor greater than 1. In a universe whose fundamental condition is an infinite state of prime and energy density which is diffused from the point of view of any and every frame of reference according to the Generalized Riemann Hypothesis, time has a forwards direction associated with the loss of prime and energy density and a backwards direction associated with a gain in prime and energy density. Because the loss of prime-density predominant over any gains in prime-density, the direction of time and of all the arrows of time is given by the GRH. There is a sense in which every change in density is either a contradiction (loss of density) or a loop (gain in density), but if and only if a change in density is outside the bounds imposed by the GRH, do these contradictions or loops take the form of a Grandfather or Shakespeare Paradox and become impossible. Collapsing the difference between the logarithmic/harmonic prime-counting functions and the primes collapses the difference between the conscious mind and the world it is conscious of, and so we obtain the superposition by taking difference between the number of primes not greater than x and an approximating formula such as

and the following wave-form:

Or from another perspective in the same phenomenon we can obtain the superposition by taking the difference between the primes themselves and an approximating formula such as

Corresponding to every crossing of the x-axis -when the difference between the subject and the object of consciousness approaches collapse- there must be either a contradiction (a self-intersecting and self-interfering event when the direction is towards a loss of prime-density) or a loop (a self-intersecting and non-self-interfering event when the direction is towards a gain in prime-density) and these self-intersections do not involve time paradoxes if and only if

The connection there is between number theory and quantum field theory can be simply illustrated (12 by associating the creation operators and to the prime numbers … Now we have identified the unique ‘factorization’ of a state into creation operators acting on the ‘vacuum’ with the unique factorization of an integer into prime numbers (and we have a hierarchy of states: |1> is the ‘vacuum’; |2> and |3> and |5> are one-particle states; |6> is a two-particle state… and so on). By reference to the Witten index (13) -the number of bosonic minus the number of fermionic zero-energy states- we see that the Mobius inversion function

={1 = n has an even number of distinct factors, -1 = n has an odd number of distinct factors, 0 = n has a repeated factor}

is equivalent to the operator that distinguishes bosonic from fermionic states, with = 0 when n has a repeated factor being equivalent to the Pauli exclusion principle. If we re-express the Mertens function (which sums the 1s and -1s of the Mobius function) as , we see that sums of these states give us essentially the same composite spiral-wave as before.

Assuming that there are an equal number of non-zero-energy bosonic and fermionic states, this wave depicts the zero-energy fluctuations of these particles, the intersections of the x-axis of which that can be identified with self-intersecting time-lines and with time-travel paradoxes of one form or the other. These examples concern the case where q =1 and a = 0, but we can easily construct approximating formulas and wave-forms for all the other possible values of q and a and observe that the self-intersections do not involve time-travel paradoxes if and only if Subscript[\[Rho], n]=1/2. This value 1/2 -associated as it is with a very particular balance of prime-density and sparsity- signifies the dividing line between classical objects that are constrained to travel in time in an arithmetic manner, and quantum objects that transcend arithmetic and are not so constrained. Infinities -other than the single infinity at root of the universe without which there is nothing- are now as it were nipped in the bud, with no need for renormalization.

Time-Travel Paradoxes on Film

Every film that involves going back in time and changing something with a view to preventing the occurrence of some future events involves the Grandfather paradox, and these films are ubiquitous – a list of books and films on the theme of time travel is a list of narratives involving the Grandfather paradox. An excellent example is The Terminator (1984):

Wikipedia summation of the plot:

In 1984 Los Angeles, a cyborg assassin known as a Terminator arrives from 2029 and steals guns and clothes. Shortly afterward, Kyle Reese, a human soldier from 2029, arrives. He steals clothes and evades the police. The Terminator begins systematically killing women named Sarah Connor, whose addresses he finds in the telephone directory. He tracks the last Sarah Connor to a nightclub, but Kyle rescues her. The pair steal a car and escape with the Terminator pursuing them in a police car.

As they hide in a parking lot, Kyle explains to Sarah that an artificial intelligence defense network, known as Skynet, will become self-aware in the near future and initiate a nuclear holocaust. Sarah’s future son John will rally the survivors and lead a resistance movement against Skynet and its army of machines. With the Resistance on the verge of victory, Skynet sent a Terminator back in time to kill Sarah before John is born, to prevent the formation of the Resistance. The Terminator is an efficient killing machine with a powerful metal endoskeleton and an external layer of living tissue that makes it appear human.

Kyle and Sarah are apprehended by police after another encounter with the Terminator. Criminal psychologist Dr. Silberman concludes that Kyle is paranoid and delusional. The Terminator repairs his body and attacks the police station, killing many police officers in his attempt to locate Sarah. Kyle and Sarah escape, steal another car and take refuge in a motel, where they assemble pipe bombs and plan their next move. Kyle admits that he has been in love with Sarah since John gave him a photograph of her, and they have sex.

The Terminator kills Sarah’s mother and impersonates her when Sarah, unaware of the Terminator’s ability to mimic victims, attempts to contact her via telephone. When they realize he has reacquired them, they escape in a pickup truck. In the ensuing chase, Kyle is wounded by gunfire while throwing pipe bombs at the Terminator. Enraged‚ Sarah knocks the Terminator off his motorcycle but loses control of the truck, which flips over. The Terminator hijacks a tank truck and attempts to run down Sarah, but Kyle slides a pipe bomb onto the tanker, causing an explosion that burns the flesh from the Terminator’s endoskeleton. It pursues them to a factory, where Kyle activates machinery to confuse the Terminator. He jams his final pipe bomb into the Terminator’s abdomen, blowing the Terminator apart, injuring Sarah, and killing Kyle. The Terminator’s still functional torso reactivates and grabs Sarah. She breaks free and lures it into a hydraulic press, crushing it.

Months later, a pregnant Sarah is traveling through Mexico, recording audio tapes to pass on to her unborn son, John. She debates whether to tell him that Kyle is his father. At a gas station, a boy takes a Polaroid photograph of her which she purchases – the same photograph that John will eventually give to Kyle.

Technically the Shakespeare paradox should always involve a repeating time loop -if someone goes back in time to give Shakespeare his plays then there is a repeating time-loop associated with the journey of the plays through time- and there are a number of films involving such time-loops including the Groundhog Day (1993):

Weatherman Phil Connors reassures Pittsburgh viewers that an approaching blizzard will miss western Pennsylvania. He goes with news producer Rita Hanson and cameraman Larry to Punxsutawney, Pennsylvania, to cover the Groundhog Day festivities. Phil makes no secret of his contempt for the assignment, the small town, and the “hicks” who live there.

The next day, Phil awakens at his Punxsutawney bed and breakfast to Sonny & Cher’s “I Got You Babe” on the clock radio. He tapes a half-hearted report on Punxsutawney Phil and the town’s festivities. Rita wants to stay and cover other events, but Phil wants to return to Pittsburgh. The blizzard blankets the region in snow, stranding them in Punxsutawney. Phil shuns the celebrations and retires to bed early.

Phil wakes to “I Got You Babe” and the same announcement from the radio, and discovers the day’s events repeating exactly. Phil relives the day and returns to bed, assuming it was a dream, but it is still Groundhog Day when he wakes: he is trapped in a time loop that no one else is aware of. Realizing there are no consequences for his actions, he spends the first several loops indulging in binge drinking, one-night stands, and reckless driving. He becomes depressed and commits suicide several times, but does not escape the loop.

Phil tries to explain his situation to Rita, for whom he has feelings, by accurately predicting the day’s events. Rita takes sympathy and they spend the entirety of one loop together, but Phil wakes up alone as usual. He decides to use his knowledge of the day’s events to better himself and the lives of others; he learns how to play the piano, sculpt ice, and speak French, but is unable to prevent the death of a homeless man.

During one loop, Phil enthusiastically reports the Groundhog Day festivities, amazing Rita. They spend the rest of the day together, with Phil impressing her with his apparent overnight transformation and charitable deeds. She successfully bids for Phil at a charity bachelor auction. Phil makes a ice sculpture of Rita’s face, and tells her that no matter what happens, even if he is doomed to continue awakening alone each morning forever, he wants her to know that he is finally happy, because he loves her. They retire together to Phil’s lodgings. Phil wakes to “I Got You Babe” again, but finds Rita is still in bed with him; he has escaped the time loop.

Based on a short story called All You Zombies by Robert A. Heinlein, a film that explores the Shakespeare paradox is the convoluted but intriguing Predestination (2014):

The film begins in medias res as a time travelling agent is trying to disarm a bomb that explodes and burns his face. Someone approaches and helps him to grasp his time travelling device, then brings him to a hospital in the future. While the agent is recovering from facial reconstruction, we learn that he has been trying to prevent the “Fizzle Bomber”‘s attack on New York, in 1975. After his recovery he receives his last assignment.

The agent moves to 1970 New York. As a bartender, he starts a conversation with one of the customers. The customer, John, writes true confession articles under the pen name “The Unmarried Mother”. This pseudonym is explained by his own life story, which he tells the bartender. The customer grew up as “Jane” in an orphanage. She excelled in her studies but had difficulty fitting in. Jane decided any children she had would be raised in a proper family, and thus avoided relationships. As an adult she applied for a program called “Space Corp”, which promised women the chance to go to space while providing astronauts R&R, but she was later disqualified because of a medical condition which she was not informed of, which a man named Robertson was interested in. Jane later met a man who said he was waiting for someone. The two fell in love with each other, but later the man disappeared. Robertson approached Jane, revealing that Space Corp worked for the Temporal Agency, and this agency now wanted to recruit her. They broke off contact when it was discovered that Jane was pregnant with her ex-lover’s baby. While performing a Caesarean section, doctors discovered she was intersex, with internalized male sex organs as well as female sex organs. Complications during the birth forced them to remove her female sex organs, and she had to undergo a gender reassignment and begin living as a man. Furthermore, the baby was stolen by a mysterious man. Since then, John has been living a bitter life as “The Unmarried Mother”.

The agent offers to take John back to the day that Jane met the lover who left her, so John can take revenge and kill him for ruining her life. In return, John will take over the agent’s job for whatever duration he wishes. The agent reveals his time travel device and the two jump to that day in 1963. John prepares to kill his past lover before he can meet Jane. While waiting, he encounters Jane, and when they begin talking, John realizes that Jane’s lover was him. The baby born from this “self-fertilization” is stolen by the agent and brought to the orphanage 18 years earlier, in 1945. Therefore, Jane, John, and their baby are the same person, revealing a predestination paradox.

The agent goes to 1975 New York, where he helps the burned man from the beginning of the film. The agent returns to 1963, a few months after he dropped John off. John has to leave Jane behind and is brought to the Temporal Agency. He now takes over the job so the agent can retire in 1975 New York, close to the day of the Fizzle Bomber’s attack. The agent’s time-travel device does not decommission itself as planned and can still be used. He has been ordered to check a launderette at the moment the Fizzle Bomber will be there. The Fizzle Bomber turns out to be the agent’s own future self, now suffering from psychosis as a result of excessive time travel. The Fizzle Bomber insists that his actions have saved and will save more lives than the lives lost, and that they ultimately lead to the reinforcement of the Temporal Agency. He tries to convince the agent that the only way to end the cycle is to spare his life, unlike the Fizzle Bomber did in his past. The agent denies he will ever become the Fizzle Bomber and kills his future self.

The film finally reveals that in 1975, John is the man who travelled to New York and was burned while disarming a bomb. His subsequent facial reconstruction significantly changes his appearance, and it is now clear that Jane, John, the agent, and the Fizzle Bomber are the same person. This agent’s creation was orchestrated by Robertson to create an agent who has no ties to time. This “perfect” temporal agent was responsible for both his own conception and death; he has driven the predestination paradox to its limit.

(10) Dirichlet, P (1837), Proof of the theorem that every unbounded arithmetic progression, whose first term and common difference are integers without common factors, contains infinitely many prime numbers